Abstract
We give a direct and rather simple construction of Martin-Löf random and rec-random sets with certain additional properties. First, reviewing the result of Gács and Kučera, given any set X we construct a Martin-Löf random set R from which X can be decoded effectively. Second, by essentially the same construction we obtain a Martin-Löf random set R that is computably enumerable selfreducible. Alternatively, using the observation that a set is computably enumerable selfreducible if and only if its associated real is computably enumerable, the existence of such a set R follows from the known fact that every Chaitin Ω real is Martin-Löf random and computably enumerable. Third, by a variant of the basic construction we obtain a rec-random set that is weak truth-table autoreducible.
The mentioned results on self- and autoreducibility complement work of Ebert, Merkle, and Vollmer [7,8,9], from which it follows that no Martin-Löf random set is Turing-autoreducible and that no rec-random set is truth-table autoreducible.
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Merkle, W., Mihailović, N. (2002). On the Construction of Effective Random Sets. In: Diks, K., Rytter, W. (eds) Mathematical Foundations of Computer Science 2002. MFCS 2002. Lecture Notes in Computer Science, vol 2420. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45687-2_47
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DOI: https://doi.org/10.1007/3-540-45687-2_47
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